# Thin-walled Pressure Vessels

A tank or pipe carrying a fluid or gas under a pressure is subjected to tensile forces, which resist bursting, developed across longitudinal and transverse sections.

**TANGENTIAL STRESS, σ _{t} (Circumferential Stress)**

Consider the tank shown being subjected to an internal pressure p. The length of the tank is L and the wall thickness is t. Isolating the right half of the tank:

The forces acting are the total pressures caused by the internal pressure p and the total tension in the walls T.

$F = pA = pDL$

$T = \sigma_t A_{wall} = \sigma_t \, tL$

$\Sigma F_H = 0$

$F = 2T$

$pDL = 2(\sigma_t \, tL)$

If there exist an external pressure p_{o} and an internal pressure p_{i}, the formula may be expressed as:

**LONGITUDINAL STRESS, σ _{L}**

Consider the free body diagram in the transverse section of the tank:

The total force acting at the rear of the tank F must equal to the total longitudinal stress on the wall P_{T} = σ_{L}A_{wall}. Since t is so small compared to D, the area of the wall is close to πDt

$F = pA = p\dfrac{\pi}{4} D^2$

$P_T = \sigma_L \pi Dt$

$\Sigma F_H = 0$

$P_T = F$

$\sigma_L \, \pi Dt = p\dfrac{\pi}{4} D^2$

If there exist an external pressure p_{o} and an internal pressure p_{i}, the formula may be expressed as:

It can be observed that the tangential stress is twice that of the longitudinal stress.

**SPHERICAL SHELL**

If a spherical tank of diameter D and thickness t contains gas under a pressure of p = p_{i} - p_{o}, the stress at the wall can be expressed as: